The rate of sedimentation is dependent upon the applied centrifugal field (G) being directed readily outwards; this is determined by the square of the angular velocity of the rotor ( in radians s-1) and the radians (r, in centimeters) of the particle from the axis of the rotation, according to the equation
Since one revolution of the rotor is equal to 2 radians, its angular velocity, in radians s-1, can be readily expressed in terms of revolutions per minute (rev min-1), the common way of expressing rotor speed being
and is generally expressed as a multiple of the earths gravitational field (g = 981 cm s-1), i.e., the ratio of the weight of the particle in the centrifugal field to the weight of the same particle when acted on by gravity alone, and is then referred to as the relative centrifugal field (RCF) or more commonly as the number times g.
When conditions for the centrifugal separation of particles are reported, therefore, rotor speed, radial dimensions and time of operation of the rotor must all be quoted. Since biochemical experiments are usually conducted with particles dissolved or suspended in solution, the rate of sedimentation of a particle is dependent not only upon the applied centrifugal field but also upon the mass of the particle, which may be expressed as the product of its volume and density, the density and viscosity of the medium in which it is sedimenting and the extent to which its shape deviates from spherical.
When particle sediments it must displace some of the solution in which it is suspended, resulting in an apparent up-thrust on the particle equal to the weight of the liquid displaced. If a particle is assumed to be spherical and of known volume and density, the latter being corrected for the buoyancy due to the density of the medium, then the net outward force (F) it experiences when centrifuged at an angular velocity of radians s-1 is given by
where 4/3r3p is the volume of a sphere of radius rp, pp is the density of the particle, pm is the density of the suspending medium, and r is the distance of the particle from the centre of rotation. Particles, however, generate friction as they migrate through the solution. If a particle is rigid and spherical and moving at a known velocity, then the frictional force (F0) opposing motion is given by
where v is the velocity or sedimentation rate of the particle, and f is the frictional coefficient of the particle in the solvent. The frictional coefficient of a particle is the function of its size, shape and hydration, and of the viscosity of the medium, and according to the Stokes equation, for an un-hydrated spherical particle, is given by
For asymmetric and/or hydrated particles, the actual radius of the particle in is replaced by the effective of Stokes radius, reff. An un-hydrated, spherical particle of known volume and density, and present in a medium of constant density, therefore accelerates in a centrifugal field, its velocity increasing until the net force of sedimentation equals the frictional force resisting its motion through the medium, i.e.,
In practice, the balancing of these forces occurs quickly and the particle reaches a constant velocity because the frictional resistance increases with the velocity of the particle. Under these conditions, the net force acting on the particle is zero. Hence, the particle no longer accelerates but achieves a maximum velocity, with the result that it now sediments at a constant rate. Its rate of sedimentation (v) is then given by
It is evident from this equation that the sedimentation rate of a given particle is proportional to its size, to the difference in density between the particle and the medium and to the applied centrifugal field. It is zero when the densities of the particle and medium are equal; it decreases when the viscosity of the medium increases, and increases as the force field increases. However, since the equation involves the square of the particle radius, it is apparent that the size of the particle has the greatest influence upon its sedimentation rate.
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For effectively removing the water-based drilling fluid filter cake and improving interfacial cementing strength and cementing quality, a new type of cementation flushing fluid (WD-C) was developed based on the strong flushing principle of water soluble fiber and the oxygenolysis principle of filter cake. It is composed of 0.5% WF-H fiber, 2.2% WF-O oxidant, 0.35% FeSO4, 1.8% KCl, 3.0% swollen powder perlite and water with its density of 1.03g/cm3. This cementation flushing fluid was systematically tested and evaluated in terms of its washing efficiency on the filter cake of water-based drilling fluid and its capacity to improve the bonding strength of cementation interface. In addition, an analysis was performed of its effect on the physical-chemical characteristics and the micro-structures of interfacial cements by means of infrared spectrum (IR), scanning electron microscope (SEM) and energy dispersive X-ray detector (EDS). It is shown that the new cementation flushing fluid presents excellent washing effect on water-based drilling fluid filter cake (with washing time within 10min). The cement particles at the cemented interface can be hydrated normally, and hydrated calcium silicate gel, Ca(OH)2 and rod-shaped ettringite (AFt) crystal are generated and interwoven with each other. In this way, dense network structures are formed, so the bonding strength of the second cementing interface rises significantly, and then cementing quality is improved. Based on the research results, one more technology is set up for removing the water-based drilling fluid filter cake efficiently and improving the bonding strength of the second cementing interface.
This paper presents the analysis, design and implementation of a model based control strategy on a copper cementation process. The cementation process is a part of the electrolyte purification section at Falconbridge Kidd Creek Division, Ontario, Canada. At Kidd Creek, zinc is produced by an aqueous electrowinning process. Before the electrowinning, the zinc solution is processed in an electrolyte purification section where impurities like copper, cobalt and cadmium are removed. In the cementation reactor, zinc dust is added to the solution to reduce the aqueous copper to metallic copper. The cementation process is in principle simple, but analysis show that there is a recognizable varying stoichiometric efficiency and a transport delay in the zinc dust that is added. The measurements are also sparse and delayed which turns this process into a challenging control problem. The cementation reactor is modelled as a dynamic CSTR and the identified time delays are incorporated in the model. This model runs on-line in parallel with the process, and an algorithm estimates the stoichiometric efficiency factor of the process. This parameter is used in a feed forward control strategy. In addition a feedback compensation is computed from either the measured or the predicted effluent copper concentration. This strategy has shown to be able to compensate for the severe disturbances in the process, and hence decrease the variability of the process.